On the local convergence of Weerakoon–Fernando method with $$\omega $$ continuity condition in Banach spaces

Abstract

In this manuscript, the study of local convergence analysis for the cubically convergent Weerakoon–Fernando method is presented to obtain solutions of nonlinear operator equations in Banach spaces. The novelty of our paper is that our analysis only requires the $$\omega $$ continuity of the first Frechet derivative and extends the applicability of the algorithm when both Lipschitz and Holder conditions fail without engaging derivatives of the higher order which do not appear on this method. The local convergence offers radii of balls of convergence, the error bounds and uniqueness of the solution. Furthermore, the convergence region for the scheme to approximate the zeros of various polynomials is studied using basins of attraction tool. Several numerical tests are performed to show the usefulness of our theoretical results.